Click here to go to the quadratic solver page; unlimited worked examples
What is the Quadratic Formula?
The quadratic formula is derived from the general form of the a quadratic equation \(ax^2+bx+c=0\). Using the coefficients \(a\), \(b\) and \(c\), we can find the roots of the equation using: \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
When to use the Quadratic Formula
It will work for any quadratic equation of the form \(ax^2+bx+c=0\) where \(a\neq0\). Sometimes
completing the square or
factorising may
be quicker if the answers are simple, but this method can be used regardless.
Click here for more information about when to use each method.
Worked Example 1: it's easy as plug and play
\[ x^2+4x+3=0 \\[6pt]a=1,\hspace{6 pt} b=4,\hspace{6 pt} c=3 \] \[\begin{align} x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\[8pt]x&=\frac{-4\pm\sqrt{4^2-4\times1\times3}}{2\times1} \\[8pt]x&=\frac{-4\pm\sqrt{16-12}}{2} \\[8pt]x&=\frac{-4\pm\sqrt{4}}{2} \\[8pt]x&=\frac{-4\pm2}{2} \\[8pt]x&=-2\pm1 \\[8pt]x&=-3\hspace{6 pt}\text{or }-1 \end{align}\]
Worked Example 2: not-so-nice values
\[ 3x^2+7x-2=0 \\[6pt]a=3,\hspace{6 pt} b=7,\hspace{6 pt} c=-2 \] \[\begin{align} x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\[8pt]x&=\frac{-7\pm\sqrt{7^2-4\times3\times(-2)}}{2\times3} \\[8pt]x&=\frac{-7\pm\sqrt{49+24}}{6} \\[8pt]x&=\frac{-7\pm\sqrt{73}}{6} \\[8pt]x&=\frac{-7+\sqrt{73}}{6} \hspace{6 pt}\text{or }\frac{-7-\sqrt{73}}{6} \end{align}\]
